2tan (problem 3.3.2)

Percentage Accurate: 42.6% → 98.8%

Time: 16.2s

Alternatives: 10

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 42.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \frac{\sin x}{\cos x}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.4e-9)
   (-
    (/ (+ (tan eps) (/ (sin x) (cos x))) (- 1.0 (* (tan x) (tan eps))))
    (tan x))
   (if (<= eps 3.2e-25)
     (+ eps (* eps (pow (tan x) 2.0)))
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (sin x) (/ (cos x) (tan eps)))))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.4e-9) {
		tmp = ((tan(eps) + (sin(x) / cos(x))) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 3.2e-25) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-4.4d-9)) then
        tmp = ((tan(eps) + (sin(x) / cos(x))) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 3.2d-25) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -4.4e-9) {
		tmp = ((Math.tan(eps) + (Math.sin(x) / Math.cos(x))) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 3.2e-25) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.sin(x) / (Math.cos(x) / Math.tan(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -4.4e-9:
		tmp = ((math.tan(eps) + (math.sin(x) / math.cos(x))) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 3.2e-25:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.sin(x) / (math.cos(x) / math.tan(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.4e-9)
		tmp = Float64(Float64(Float64(tan(eps) + Float64(sin(x) / cos(x))) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 3.2e-25)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -4.4e-9)
		tmp = ((tan(eps) + (sin(x) / cos(x))) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 3.2e-25)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.4e-9], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-25], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.3999999999999997e-9

   1. Initial program 58.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.4%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.4%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 60.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x + \tan \varepsilon\right)\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   7. Applied egg-rr 60.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x + \tan \varepsilon\right)\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
   8. Step-by-step derivation

      1. expm1-log1p-u 99.4%

         \[\leadsto \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

      2. tan-quot 99.5%

         \[\leadsto \frac{\color{blue}{\frac{\sin x}{\cos x}} + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

      3. div-inv 99.4%

         \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{1}{\cos x}} + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

      4. fma-def 99.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, \tan \varepsilon\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   9. Applied egg-rr 99.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, \tan \varepsilon\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
   10. Step-by-step derivation

       1. fma-udef 99.4%

          \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{1}{\cos x} + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

       2. associate-*r/ 99.5%

          \[\leadsto \frac{\color{blue}{\frac{\sin x \cdot 1}{\cos x}} + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

       3. *-rgt-identity 99.5%

          \[\leadsto \frac{\frac{\color{blue}{\sin x}}{\cos x} + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   11. Simplified 99.5%

       \[\leadsto \frac{\color{blue}{\frac{\sin x}{\cos x} + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   if -4.3999999999999997e-9 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]

   if 3.2000000000000001e-25 < eps

   1. Initial program 66.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.6%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.5%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.5%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   6. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}} - \tan x \]

      2. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x} - \tan x \]

      3. clear-num 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \cdot \tan x} - \tan x \]

      4. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x \]

      5. frac-times 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1 \cdot \sin x}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}}} - \tan x \]

      6. *-un-lft-identity 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x}}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}} - \tan x \]

      7. clear-num 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \cos x}} - \tan x \]

      8. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{1}{\color{blue}{\tan \varepsilon}} \cdot \cos x}} - \tan x \]

   7. Applied egg-rr 99.6%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\cos x \cdot \frac{1}{\tan \varepsilon}}}} - \tan x \]

      2. associate-*r/ 99.7%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{\cos x \cdot 1}{\tan \varepsilon}}}} - \tan x \]

      3. *-rgt-identity 99.7%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\color{blue}{\cos x}}{\tan \varepsilon}}} - \tan x \]

   9. Simplified 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \frac{\sin x}{\cos x}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ \frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \sin x} + t_0}{\frac{1 - t_0}{\tan x}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))))
   (/
    (+ (/ (* (cos x) (sin eps)) (* (cos eps) (sin x))) t_0)
    (/ (- 1.0 t_0) (tan x)))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	return (((cos(x) * sin(eps)) / (cos(eps) * sin(x))) + t_0) / ((1.0 - t_0) / tan(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(x) * tan(eps)
    code = (((cos(x) * sin(eps)) / (cos(eps) * sin(x))) + t_0) / ((1.0d0 - t_0) / tan(x))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) * Math.tan(eps);
	return (((Math.cos(x) * Math.sin(eps)) / (Math.cos(eps) * Math.sin(x))) + t_0) / ((1.0 - t_0) / Math.tan(x));
}

Python

def code(x, eps):
	t_0 = math.tan(x) * math.tan(eps)
	return (((math.cos(x) * math.sin(eps)) / (math.cos(eps) * math.sin(x))) + t_0) / ((1.0 - t_0) / math.tan(x))

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	return Float64(Float64(Float64(Float64(cos(x) * sin(eps)) / Float64(cos(eps) * sin(x))) + t_0) / Float64(Float64(1.0 - t_0) / tan(x)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) * tan(eps);
	tmp = (((cos(x) * sin(eps)) / (cos(eps) * sin(x))) + t_0) / ((1.0 - t_0) / tan(x));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 44.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Step-by-step derivation

   1. tan-sum 64.9%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 64.9%

      \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   3. *-un-lft-identity 64.9%

      \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

   4. prod-diff 64.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

   5. *-commutative 64.9%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

   6. *-un-lft-identity 64.9%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

   7. *-commutative 64.9%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

   8. *-un-lft-identity 64.9%

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

3. Applied egg-rr 64.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
4. Step-by-step derivation

   1. +-commutative 64.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   2. fma-udef 64.9%

      \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

   3. associate-+r+ 64.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

   4. unsub-neg 64.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

5. Simplified 64.9%

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
6. Step-by-step derivation

   1. *-commutative 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}} - \tan x \]

   2. tan-quot 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x} - \tan x \]

   3. clear-num 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \cdot \tan x} - \tan x \]

   4. tan-quot 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x \]

   5. frac-times 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1 \cdot \sin x}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}}} - \tan x \]

   6. *-un-lft-identity 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x}}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}} - \tan x \]

   7. clear-num 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \cos x}} - \tan x \]

   8. tan-quot 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{1}{\color{blue}{\tan \varepsilon}} \cdot \cos x}} - \tan x \]

7. Applied egg-rr 64.9%

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
8. Step-by-step derivation

   1. *-commutative 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\cos x \cdot \frac{1}{\tan \varepsilon}}}} - \tan x \]

   2. *-rgt-identity 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x \cdot 1}}{\cos x \cdot \frac{1}{\tan \varepsilon}}} - \tan x \]

   3. times-frac 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \frac{1}{\frac{1}{\tan \varepsilon}}}} - \tan x \]

   4. remove-double-div 64.9%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\cos x} \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

9. Simplified 64.9%

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \tan \varepsilon}} - \tan x \]
10. Step-by-step derivation

    1. tan-quot 64.9%

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \tan \varepsilon} - \tan x \]

    2. tan-quot 64.8%

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}} \]

    3. clear-num 64.8%

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{1}{\frac{\cos x}{\sin x}}} \]

    4. frac-sub 64.5%

       \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}}} \]

    5. clear-num 64.5%

       \[\leadsto \frac{\left(\tan x + \tan \varepsilon\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{\cos x}}} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}} \]

    6. tan-quot 64.7%

       \[\leadsto \frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\color{blue}{\tan x}} - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}} \]

    7. *-commutative 64.7%

       \[\leadsto \frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} - \color{blue}{1 \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}} \]

    8. *-un-lft-identity 64.7%

       \[\leadsto \frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} - \color{blue}{\left(1 - \tan x \cdot \tan \varepsilon\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{\cos x}{\sin x}} \]

    9. clear-num 64.7%

       \[\leadsto \frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} - \left(1 - \tan x \cdot \tan \varepsilon\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \color{blue}{\frac{1}{\frac{\sin x}{\cos x}}}} \]

    10. tan-quot 64.7%

        \[\leadsto \frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} - \left(1 - \tan x \cdot \tan \varepsilon\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\color{blue}{\tan x}}} \]

11. Applied egg-rr 64.7%

    \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} - \left(1 - \tan x \cdot \tan \varepsilon\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}}} \]
12. Step-by-step derivation

    1. associate--r- 67.9%

       \[\leadsto \frac{\color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} - 1\right) + \tan x \cdot \tan \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}} \]

    2. sub-neg 67.9%

       \[\leadsto \frac{\color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{\tan x} + \left(-1\right)\right)} + \tan x \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}} \]

    3. associate-*r/ 68.4%

       \[\leadsto \frac{\left(\color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{\tan x}} + \left(-1\right)\right) + \tan x \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}} \]

    4. *-rgt-identity 68.4%

       \[\leadsto \frac{\left(\frac{\color{blue}{\tan x + \tan \varepsilon}}{\tan x} + \left(-1\right)\right) + \tan x \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}} \]

    5. metadata-eval 68.4%

       \[\leadsto \frac{\left(\frac{\tan x + \tan \varepsilon}{\tan x} + \color{blue}{-1}\right) + \tan x \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{1}{\tan x}} \]

    6. associate-*r/ 68.4%

       \[\leadsto \frac{\left(\frac{\tan x + \tan \varepsilon}{\tan x} + -1\right) + \tan x \cdot \tan \varepsilon}{\color{blue}{\frac{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot 1}{\tan x}}} \]

    7. *-rgt-identity 68.4%

       \[\leadsto \frac{\left(\frac{\tan x + \tan \varepsilon}{\tan x} + -1\right) + \tan x \cdot \tan \varepsilon}{\frac{\color{blue}{1 - \tan x \cdot \tan \varepsilon}}{\tan x}} \]

13. Simplified 68.4%

    \[\leadsto \color{blue}{\frac{\left(\frac{\tan x + \tan \varepsilon}{\tan x} + -1\right) + \tan x \cdot \tan \varepsilon}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x}}} \]

14. Taylor expanded in x around inf 99.1%

    \[\leadsto \frac{\color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \sin x}} + \tan x \cdot \tan \varepsilon}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x}} \]

15. Final simplification 99.1%

    \[\leadsto \frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \sin x} + \tan x \cdot \tan \varepsilon}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x}} \]

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2.9e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 3.2e-25)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (/ t_0 (- 1.0 (/ (sin x) (/ (cos x) (tan eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.9e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 3.2e-25) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-2.9d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 3.2d-25) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 / (1.0d0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -2.9e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 3.2e-25) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (Math.sin(x) / (Math.cos(x) / Math.tan(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -2.9e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 3.2e-25:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 / (1.0 - (math.sin(x) / (math.cos(x) / math.tan(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.9e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 3.2e-25)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -2.9e-9)
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 3.2e-25)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.9e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-25], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.89999999999999991e-9

   1. Initial program 58.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.4%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.4%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]

   if -2.89999999999999991e-9 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]

   if 3.2000000000000001e-25 < eps

   1. Initial program 66.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.6%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.5%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.5%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   6. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}} - \tan x \]

      2. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x} - \tan x \]

      3. clear-num 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \cdot \tan x} - \tan x \]

      4. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x \]

      5. frac-times 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1 \cdot \sin x}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}}} - \tan x \]

      6. *-un-lft-identity 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x}}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}} - \tan x \]

      7. clear-num 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \cos x}} - \tan x \]

      8. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{1}{\color{blue}{\tan \varepsilon}} \cdot \cos x}} - \tan x \]

   7. Applied egg-rr 99.6%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\cos x \cdot \frac{1}{\tan \varepsilon}}}} - \tan x \]

      2. associate-*r/ 99.7%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{\cos x \cdot 1}{\tan \varepsilon}}}} - \tan x \]

      3. *-rgt-identity 99.7%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\color{blue}{\cos x}}{\tan \varepsilon}}} - \tan x \]

   9. Simplified 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.45e-9)
     (- (/ t_0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) (tan x))
     (if (<= eps 3.2e-25)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (/ t_0 (- 1.0 (/ (sin x) (/ (cos x) (tan eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.45e-9) {
		tmp = (t_0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	} else if (eps <= 3.2e-25) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-3.45d-9)) then
        tmp = (t_0 / (1.0d0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x)
    else if (eps <= 3.2d-25) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 / (1.0d0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -3.45e-9) {
		tmp = (t_0 / (1.0 - ((Math.sin(x) * Math.tan(eps)) / Math.cos(x)))) - Math.tan(x);
	} else if (eps <= 3.2e-25) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (Math.sin(x) / (Math.cos(x) / Math.tan(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -3.45e-9:
		tmp = (t_0 / (1.0 - ((math.sin(x) * math.tan(eps)) / math.cos(x)))) - math.tan(x)
	elif eps <= 3.2e-25:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 / (1.0 - (math.sin(x) / (math.cos(x) / math.tan(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.45e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) - tan(x));
	elseif (eps <= 3.2e-25)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -3.45e-9)
		tmp = (t_0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	elseif (eps <= 3.2e-25)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.45e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-25], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.44999999999999987e-9

   1. Initial program 58.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.4%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.4%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.3%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   6. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}} - \tan x \]

      2. tan-quot 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x} - \tan x \]

      3. clear-num 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \cdot \tan x} - \tan x \]

      4. tan-quot 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x \]

      5. frac-times 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1 \cdot \sin x}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}}} - \tan x \]

      6. *-un-lft-identity 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x}}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}} - \tan x \]

      7. clear-num 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \cos x}} - \tan x \]

      8. tan-quot 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{1}{\color{blue}{\tan \varepsilon}} \cdot \cos x}} - \tan x \]

   7. Applied egg-rr 99.4%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\cos x \cdot \frac{1}{\tan \varepsilon}}}} - \tan x \]

      2. *-rgt-identity 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x \cdot 1}}{\cos x \cdot \frac{1}{\tan \varepsilon}}} - \tan x \]

      3. times-frac 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \frac{1}{\frac{1}{\tan \varepsilon}}}} - \tan x \]

      4. remove-double-div 99.4%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\cos x} \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   9. Simplified 99.4%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \tan \varepsilon}} - \tan x \]
   10. Step-by-step derivation

       1. associate-*l/ 99.5%

          \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x \]

       2. *-commutative 99.5%

          \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\tan \varepsilon \cdot \sin x}}{\cos x}} - \tan x \]

   11. Applied egg-rr 99.5%

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan \varepsilon \cdot \sin x}{\cos x}}} - \tan x \]

   if -3.44999999999999987e-9 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]

   if 3.2000000000000001e-25 < eps

   1. Initial program 66.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.6%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.5%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.5%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.5%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.5%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   6. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}} - \tan x \]

      2. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x} - \tan x \]

      3. clear-num 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \cdot \tan x} - \tan x \]

      4. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x \]

      5. frac-times 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1 \cdot \sin x}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}}} - \tan x \]

      6. *-un-lft-identity 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x}}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}} - \tan x \]

      7. clear-num 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \cos x}} - \tan x \]

      8. tan-quot 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{1}{\color{blue}{\tan \varepsilon}} \cdot \cos x}} - \tan x \]

   7. Applied egg-rr 99.6%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\cos x \cdot \frac{1}{\tan \varepsilon}}}} - \tan x \]

      2. associate-*r/ 99.7%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{\cos x \cdot 1}{\tan \varepsilon}}}} - \tan x \]

      3. *-rgt-identity 99.7%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\color{blue}{\cos x}}{\tan \varepsilon}}} - \tan x \]

   9. Simplified 99.7%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 5: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.6e-9) (not (<= eps 3.2e-25)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.6e-9) || !(eps <= 3.2e-25)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.6d-9)) .or. (.not. (eps <= 3.2d-25))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.6e-9) || !(eps <= 3.2e-25)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.6e-9) or not (eps <= 3.2e-25):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.6e-9) || !(eps <= 3.2e-25))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.6e-9) || ~((eps <= 3.2e-25)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.6e-9], N[Not[LessEqual[eps, 3.2e-25]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.6e-9 or 3.2000000000000001e-25 < eps

   1. Initial program 61.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.5%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.4%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]

   if -3.6e-9 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 6: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.7e-6) (not (<= eps 3.2e-25)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* x (tan eps)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.7e-6) || !(eps <= 3.2e-25)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (x * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.7d-6)) .or. (.not. (eps <= 3.2d-25))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (x * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.7e-6) || !(eps <= 3.2e-25)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (x * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.7e-6) or not (eps <= 3.2e-25):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (x * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.7e-6) || !(eps <= 3.2e-25))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(x * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.7e-6) || ~((eps <= 3.2e-25)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (x * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.7e-6], N[Not[LessEqual[eps, 3.2e-25]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(x * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.69999999999999989e-6 or 3.2000000000000001e-25 < eps

   1. Initial program 61.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Step-by-step derivation

      1. tan-sum 99.5%

         \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      2. div-inv 99.4%

         \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

      3. *-un-lft-identity 99.4%

         \[\leadsto \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{1 \cdot \tan x} \]

      4. prod-diff 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x \cdot 1\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right)} \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{1 \cdot \tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      6. *-un-lft-identity 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\color{blue}{\tan x}\right) + \mathsf{fma}\left(-\tan x, 1, \tan x \cdot 1\right) \]

      7. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{1 \cdot \tan x}\right) \]

      8. *-un-lft-identity 99.4%

         \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \color{blue}{\tan x}\right) \]

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

      2. fma-udef 99.4%

         \[\leadsto \mathsf{fma}\left(-\tan x, 1, \tan x\right) + \color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]

      3. associate-+r+ 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right)} \]

      4. unsub-neg 99.4%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-\tan x, 1, \tan x\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) - \tan x} \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
   6. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}} - \tan x \]

      2. tan-quot 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \tan x} - \tan x \]

      3. clear-num 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}} \cdot \tan x} - \tan x \]

      4. tan-quot 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} \cdot \color{blue}{\frac{\sin x}{\cos x}}} - \tan x \]

      5. frac-times 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{1 \cdot \sin x}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}}} - \tan x \]

      6. *-un-lft-identity 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x}}{\frac{\cos \varepsilon}{\sin \varepsilon} \cdot \cos x}} - \tan x \]

      7. clear-num 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \cos x}} - \tan x \]

      8. tan-quot 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{1}{\color{blue}{\tan \varepsilon}} \cdot \cos x}} - \tan x \]

   7. Applied egg-rr 99.5%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\frac{1}{\tan \varepsilon} \cdot \cos x}}} - \tan x \]
   8. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\color{blue}{\cos x \cdot \frac{1}{\tan \varepsilon}}}} - \tan x \]

      2. *-rgt-identity 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin x \cdot 1}}{\cos x \cdot \frac{1}{\tan \varepsilon}}} - \tan x \]

      3. times-frac 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \frac{1}{\frac{1}{\tan \varepsilon}}}} - \tan x \]

      4. remove-double-div 99.5%

         \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\cos x} \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   9. Simplified 99.5%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x} \cdot \tan \varepsilon}} - \tan x \]

   10. Taylor expanded in x around 0 64.5%

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{x} \cdot \tan \varepsilon} - \tan x \]

   if -4.69999999999999989e-6 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.7 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 7: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.2e-6) (not (<= eps 3.2e-25)))
   (tan eps)
   (* eps (+ 1.0 (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.2e-6) || !(eps <= 3.2e-25)) {
		tmp = tan(eps);
	} else {
		tmp = eps * (1.0 + pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-8.2d-6)) .or. (.not. (eps <= 3.2d-25))) then
        tmp = tan(eps)
    else
        tmp = eps * (1.0d0 + (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.2e-6) || !(eps <= 3.2e-25)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * (1.0 + Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -8.2e-6) or not (eps <= 3.2e-25):
		tmp = math.tan(eps)
	else:
		tmp = eps * (1.0 + math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.2e-6) || !(eps <= 3.2e-25))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(1.0 + (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -8.2e-6) || ~((eps <= 3.2e-25)))
		tmp = tan(eps);
	else
		tmp = eps * (1.0 + (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.2e-6], N[Not[LessEqual[eps, 3.2e-25]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.1999999999999994e-6 or 3.2000000000000001e-25 < eps

   1. Initial program 61.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in x around 0 63.8%

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
   3. Step-by-step derivation

      1. tan-quot 64.0%

         \[\leadsto \color{blue}{\tan \varepsilon} \]

      2. expm1-log1p-u 48.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

      3. expm1-udef 45.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]

   4. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 48.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

      2. expm1-log1p 64.0%

         \[\leadsto \color{blue}{\tan \varepsilon} \]

   6. Simplified 64.0%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

   if -8.1999999999999994e-6 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \varepsilon + \color{blue}{\varepsilon \cdot {\tan x}^{2}} \]

      2. *-rgt-identity 99.6%

         \[\leadsto \color{blue}{\varepsilon \cdot 1} + \varepsilon \cdot {\tan x}^{2} \]

      3. distribute-lft-in 99.5%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + {\tan x}^{2}\right)} \]

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + {\tan x}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \]

Alternative 8: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.9 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.9e-7) (not (<= eps 3.2e-25)))
   (tan eps)
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.9e-7) || !(eps <= 3.2e-25)) {
		tmp = tan(eps);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-8.9d-7)) .or. (.not. (eps <= 3.2d-25))) then
        tmp = tan(eps)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.9e-7) || !(eps <= 3.2e-25)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -8.9e-7) or not (eps <= 3.2e-25):
		tmp = math.tan(eps)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.9e-7) || !(eps <= 3.2e-25))
		tmp = tan(eps);
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -8.9e-7) || ~((eps <= 3.2e-25)))
		tmp = tan(eps);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.9e-7], N[Not[LessEqual[eps, 3.2e-25]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.899999999999999e-7 or 3.2000000000000001e-25 < eps

   1. Initial program 61.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in x around 0 63.8%

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
   3. Step-by-step derivation

      1. tan-quot 64.0%

         \[\leadsto \color{blue}{\tan \varepsilon} \]

      2. expm1-log1p-u 48.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

      3. expm1-udef 45.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]

   4. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 48.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

      2. expm1-log1p 64.0%

         \[\leadsto \color{blue}{\tan \varepsilon} \]

   6. Simplified 64.0%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

   if -8.899999999999999e-7 < eps < 3.2000000000000001e-25

   1. Initial program 25.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   3. Step-by-step derivation

      1. cancel-sign-sub-inv 99.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

      3. *-lft-identity 99.5%

         \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 99.5%

         \[\leadsto \color{blue}{1 \cdot \varepsilon + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon} \]

      2. *-un-lft-identity 99.5%

         \[\leadsto \color{blue}{\varepsilon} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon \]

      3. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}} \cdot \varepsilon \]

      4. unpow2 99.5%

         \[\leadsto \varepsilon + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}} \cdot \varepsilon \]

      5. frac-times 99.5%

         \[\leadsto \varepsilon + \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} \cdot \varepsilon \]

      6. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}\right) \cdot \varepsilon \]

      7. tan-quot 99.6%

         \[\leadsto \varepsilon + \left(\tan x \cdot \color{blue}{\tan x}\right) \cdot \varepsilon \]

      8. pow2 99.6%

         \[\leadsto \varepsilon + \color{blue}{{\tan x}^{2}} \cdot \varepsilon \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon + {\tan x}^{2} \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.9 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 9: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (tan eps))

C

double code(double x, double eps) {
	return tan(eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan(eps)
end function

Java

public static double code(double x, double eps) {
	return Math.tan(eps);
}

Python

def code(x, eps):
	return math.tan(eps)

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = tan(eps);
end

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 44.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in x around 0 59.6%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
3. Step-by-step derivation

   1. tan-quot 59.8%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

   2. expm1-log1p-u 51.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

   3. expm1-udef 27.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]

4. Applied egg-rr 27.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-def 51.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \varepsilon\right)\right)} \]

   2. expm1-log1p 59.8%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

6. Simplified 59.8%

   \[\leadsto \color{blue}{\tan \varepsilon} \]

7. Final simplification 59.8%

   \[\leadsto \tan \varepsilon \]

Alternative 10: 30.7% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 44.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in x around 0 59.6%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

3. Taylor expanded in eps around 0 29.2%

   \[\leadsto \color{blue}{\varepsilon} \]

4. Final simplification 29.2%

   \[\leadsto \varepsilon \]

Developer target: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (cos(x) * cos((x + eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (Math.cos(x) * Math.cos((x + eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023320 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))